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Mirrors > Home > MPE Home > Th. List > chpmatply1 | Structured version Visualization version GIF version |
Description: The characteristic polynomial of a (square) matrix over a commutative ring is a polynomial, see also the following remark in [Lang], p. 561: "[the characteristic polynomial] is an element of k[t]". (Contributed by AV, 2-Aug-2019.) (Proof shortened by AV, 29-Nov-2019.) |
Ref | Expression |
---|---|
chpmatply1.c | ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) |
chpmatply1.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
chpmatply1.b | ⊢ 𝐵 = (Base‘𝐴) |
chpmatply1.p | ⊢ 𝑃 = (Poly1‘𝑅) |
chpmatply1.e | ⊢ 𝐸 = (Base‘𝑃) |
Ref | Expression |
---|---|
chpmatply1 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chpmatply1.c | . . 3 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) | |
2 | chpmatply1.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | chpmatply1.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
4 | chpmatply1.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | eqid 2819 | . . 3 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃) | |
6 | eqid 2819 | . . 3 ⊢ (𝑁 maDet 𝑃) = (𝑁 maDet 𝑃) | |
7 | eqid 2819 | . . 3 ⊢ (-g‘(𝑁 Mat 𝑃)) = (-g‘(𝑁 Mat 𝑃)) | |
8 | eqid 2819 | . . 3 ⊢ (var1‘𝑅) = (var1‘𝑅) | |
9 | eqid 2819 | . . 3 ⊢ ( ·𝑠 ‘(𝑁 Mat 𝑃)) = ( ·𝑠 ‘(𝑁 Mat 𝑃)) | |
10 | eqid 2819 | . . 3 ⊢ (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅) | |
11 | eqid 2819 | . . 3 ⊢ (1r‘(𝑁 Mat 𝑃)) = (1r‘(𝑁 Mat 𝑃)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | chpmatval 21431 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = ((𝑁 maDet 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)))) |
13 | 4 | ply1crng 20358 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
14 | 13 | 3ad2ant2 1129 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ CRing) |
15 | crngring 19300 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
16 | eqid 2819 | . . . . 5 ⊢ (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) = (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) | |
17 | 2, 3, 4, 5, 8, 10, 7, 9, 11, 16 | chmatcl 21428 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) ∈ (Base‘(𝑁 Mat 𝑃))) |
18 | 15, 17 | syl3an2 1159 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) ∈ (Base‘(𝑁 Mat 𝑃))) |
19 | eqid 2819 | . . . 4 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃)) | |
20 | chpmatply1.e | . . . 4 ⊢ 𝐸 = (Base‘𝑃) | |
21 | 6, 5, 19, 20 | mdetcl 21197 | . . 3 ⊢ ((𝑃 ∈ CRing ∧ (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) ∈ (Base‘(𝑁 Mat 𝑃))) → ((𝑁 maDet 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀))) ∈ 𝐸) |
22 | 14, 18, 21 | syl2anc 586 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 maDet 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀))) ∈ 𝐸) |
23 | 12, 22 | eqeltrd 2911 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 Fincfn 8501 Basecbs 16475 ·𝑠 cvsca 16561 -gcsg 18097 1rcur 19243 Ringcrg 19289 CRingccrg 19290 var1cv1 20336 Poly1cpl1 20337 Mat cmat 21008 maDet cmdat 21185 matToPolyMat cmat2pmat 21304 CharPlyMat cchpmat 21426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-addf 10608 ax-mulf 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-xor 1499 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-ot 4568 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-ofr 7402 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-tpos 7884 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-sup 8898 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-xnn0 11960 df-z 11974 df-dec 12091 df-uz 12236 df-rp 12382 df-fz 12885 df-fzo 13026 df-seq 13362 df-exp 13422 df-hash 13683 df-word 13854 df-lsw 13907 df-concat 13915 df-s1 13942 df-substr 13995 df-pfx 14025 df-splice 14104 df-reverse 14113 df-s2 14202 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-efmnd 18026 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-gim 18391 df-cntz 18439 df-oppg 18466 df-symg 18488 df-pmtr 18562 df-psgn 18611 df-cmn 18900 df-abl 18901 df-mgp 19232 df-ur 19244 df-ring 19291 df-cring 19292 df-oppr 19365 df-dvdsr 19383 df-unit 19384 df-invr 19414 df-dvr 19425 df-rnghom 19459 df-drng 19496 df-subrg 19525 df-lmod 19628 df-lss 19696 df-sra 19936 df-rgmod 19937 df-ascl 20079 df-psr 20128 df-mvr 20129 df-mpl 20130 df-opsr 20132 df-psr1 20340 df-vr1 20341 df-ply1 20342 df-cnfld 20538 df-zring 20610 df-zrh 20643 df-dsmm 20868 df-frlm 20883 df-mamu 20987 df-mat 21009 df-mdet 21186 df-mat2pmat 21307 df-chpmat 21427 |
This theorem is referenced by: chmaidscmat 21448 cpmidgsum 21468 cpmidgsumm2pm 21469 cpmidpmatlem2 21471 cpmidpmatlem3 21472 chcoeffeqlem 21485 cayhamlem3 21487 cayleyhamilton1 21492 |
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